UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLV
2007
PLURISUBHARMONIC EXTENSION IN NON-DEGENERATE
ANALYTIC POLYHEDRA
by Per Åhag, Rafal Czyż, Sam Lodin and Frank Wikström
Abstract. Let P be a connected, non-degenerate analytic polyhedron in
Cn , n ≥ 2. In this note we characterize those continuous functions ∂P → R
which can be extended to a plurisubharmonic function in P.
1. Introduction. Let D be a bounded, connected and open set in Cn ,
n ≥ 1, and let f1 , f2 , . . . , fN , (N ≥ n) be holomorphic on D. The set
P = {z ∈ D : |fi (z)| < 1, i = 1, . . . , N } ,
is called an analytic polyhedron if P is relatively compact in D. The functions
fi will be called defining functions for P. For every increasing multi-index I,
1 ≤ i1 < · · · < ik ≤ N , of length k ∈ {1, . . . , n} let
σI = {z ∈ Cn : |fi (z)| = 1, i ∈ I} ∩ P̄ .
The distinguished boundary of an analytic polyhedron is the part of the topological boundary that is given by
[
Γn =
σI .
|I|=n
In this note we will assume, in addition, that the analytic polyhedron P is
non-degenerate, i.e., for every increasing multi-index 1 ≤ i1 < · · · < ik ≤ N of
length k ∈ {1, . . . , n} we have that ∂fi1 ∧ · · · ∧ ∂fik 6= 0 on σI . In particular,
this assumption implies that σI is an orientable (n − k)-dimensional complex
2000 Mathematics Subject Classification. Primary 31C10; Secondary 32U15.
Key words and phrases. Analytic disc, analytic polyhedra, Dirichlet problem, Jensen
measure, plurisubharmonic function.
The second-named author was partially supported by KBN, under grant number N N201
3679 33.
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submanifold of Cn with the natural orientation induced by Cn . A polyhedron
is not necessarily connected, but in this setting it has at most a finite number
of connected components (see e.g. [1]).
Throughout this note, we assume that P is a connected, non-degenerate
analytic polyhedron in Cn , n ≥ 2. As in [5], consider the bounded plurisubharmonic function ϕ : P → (−∞, 0) defined by
ϕ(z) = max{log |f1 (z)|, . . . , log |fN (z)|, −1}.
The set {z ∈ P : ϕ(z) < c} is relatively compact in P for every c ∈ (−∞, 0)
and therefore P is a so called hyperconvex domain. Recall that a hyperconvex
domain is regular with respect to the Laplace operator, i.e., if Ω is hyperconvex, then every continuous real-valued function defined on ∂Ω has a harmonic
extension to Ω and this extension can be taken continuous on Ω̄ (see e.g. [7]).
Given a continuous, real-valued function f on ∂P, it is not always possible
to find a plurisubharmonic extension of f . However, if f has a plurisubharmonic extension, then it is possible to solve certain non-homogenous complex
Monge–Ampère equations (see [2, 3]). On the other hand, if we prescribe
the function on a smaller part of the topological boundary, more precisely on
Γn , it was proved in [5] that we can always find a plurisubharmonic function with essentially the given boundary values. This was later generalized by
Górski [10] using the same techniques. Bremermann’s result in [5] was also the
starting point in Levenberg and Okada [12], where the authors consider the
non-homogenous complex Monge–Ampère equation in the bidisc. In this note
our main goal is to characterize those continuous functions ∂P → R which can
be extended to a plurisubharmonic function in P. Our main result is:
Theorem A. Assume that P is a connected, non-degenerate analytic polyhedron in Cn , n ≥ 2. If f : ∂P → R is a real-valued continuous function, then
the following assertions are equivalent:
(1) there exists a function s which is plurisubharmonic on P, continuous on
P̄ and s|∂P = f ,
(2) for every increasing multi-index I of length k ∈ {1, . . . , n − 1} the function
f is plurisubharmonic on the (n − k)-dimensional complex manifold σI ,
(3) the function f is subharmonic on every analytic disc d embedded in ∂P,
i.e., the function f ◦ d is subharmonic on the unit disc ∆ in C for every
injective, holomorphic function d : ∆ → P̄ with d(∆) ⊆ ∂P,
Using well-known results we achieve that the above conditions are also
equivalent to:
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(4) for every p ∈ ∂P and every Jensen measure µ, defined on P̄ with
barycenter p it holds that
Z
f (p) ≤
f dµ
∂P
(see Section 2 for the definition of Jensen measures),
(5) the Perron–Bremermann envelope P Bf is a maximal plurisubharmonic
function on P, continuous on P̄, and limz→ξ P Bf (z) = f (ξ) for every
ξ ∈ ∂P (see Section 2 for the definition of the Perron–Bremermann
envelope).
The equivalence between (1) and (4) was proved by the fourth-named author in [15] and the equivalence between (1) and (5) follows by Walsh’s theorem
(see Section 2 or [14]). For the case n = 1, properties (2) and (3) do not make
sense and (1), (4) and (5) are valid for every continuous function on ∂P. If
n ≥ 2, then the unit polydisc ∆n ⊆ Cn , is an example of a connected, nondegenerate analytic polyhedron given by the n coordinate functions as defining
functions. In the polydisc case the equivalence between property (1) and (3)
was shown by Sadullaev [13] for n = 2 and later this was generalized by Blocki
in [4] to n ≥ 2 (see also Example 3.6 in [15]).
2. The boundaries of a polyhedra and preliminaries. In this section
we will first recall some different notions of boundaries. Let P be an analytic
polyhedron in Cn , n ≥ 1, without any extra assumption. The topological
boundary ∂P of the polyhedron is given by
[
∂P =
σI ,
|I|=k
1≤k≤n
where σI is defined as in the introduction by
σI = {z ∈ Cn : |fi (z)| = 1, i ∈ I} ∩ P̄ .
Let A(P ) denote the algebra of continuous functions on P̄ that are holomorphic
on P. The Šilov boundary with respect to the algebra A(P ) is defined as the
smallest closed set S(P ) ⊆ P̄ with the property that the modulus of any
function in A(P ) attains its maximum on S(P ). The distinguished boundary
is in general larger than the Šilov boundary with respect to A(P ). It was proved
in [11] that under the assumption that P is non-degenerate, the distinguished
boundary and Šilov boundary with respect to A(P ) are the same. Moreover,
they coincide with the set of points in ∂P through which there is no germ
of an analytic variety of positive dimension contained in ∂P. From these
considerations, Proposition 2.1 follows. We refer to [9] for more information
about the function algebraic viewpoint of boundaries.
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Proposition 2.1. Assume that P is a non-degenerate analytic polyhedron
in Cn , n ≥ 2. Let ∆ ⊆ C be the open unit disc and assume that d : ∆ → P̄ is
a holomorphic function, then the following holds:
a) if d(∆) ⊆ Γn , then d is constant.
b) if d(∆) ⊆ ∂P is non-constant, then there exists an increasing multiindex I of length k ∈ {1, . . . , n}, such that d(∆) ⊆ σI .
Remark. Without the assumption of non-degeneracy of P, there can be
analytic structure in the Šilov boundary (see e.g. [11]).
Definition 2.2. Let Ω ⊆ Cn be a bounded domain and let µ be a nonnegative, regular Borel measure supported on Ω̄. The measure µ is a Jensen
measure with barycenter at z ∈ Ω̄ if
Z
u(z) ≤
u dµ ,
Ω̄
for every continuous function u : Ω̄ → [−∞, ∞), not identically −∞, such that
u ∈ PSH(Ω). The set of all Jensen measures with barycenter at z will be
denoted by Jzc (Ω̄).
Definition 2.2 is a slightly different definition of Jensen measures than the
classical one, since it allows the measure to have support in Ω̄, and also since
it considers Jensen measures for boundary points. It was proved in [15] (see
also [6]) that if Ω is a bounded hyperconvex domain and µ ∈ Jzc (Ω̄), z ∈ ∂Ω,
then the support of µ is contained in ∂Ω. In the case of an analytic polyhedron
we can gain more detailed information on the support of a Jensen measure with
barycenter at a boundary point.
Proposition 2.3. Let P be an analytic polyhedron in Cn , n ≥ 2, and
let p ∈ ∂Ω. Define J = {j ∈ I : |fj (p)| = 1} and Σp = ∩j∈J fj−1 (fj (p)).
If µ ∈ Jpc (P̄), then the support of µ is contained in Σp and µ ∈ Jzc (Σ̄p ).
Conversely, if µ is a probability measure defined on P with support in Σ̄p such
that µ ∈ Jpc (Σ̄p ), then µ ∈ Jpc (P̄).
Proof. Let u be a function defined on P̄ by
X
u(z) =
(|fj (z) + fj (p)| − 2) .
j∈J
Then u ∈ PSH(P) ∩ C(P̄), u ≤ 0 on P̄, and u(z) = 0 exactly for z ∈ Σp .
Hence, if µ ∈ Jpc (P̄), then
Z
0 = u(p) ≤
u dµ ≤ 0 ,
P̄
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which implies that u = 0 almost everywhere on P̄ with respect to µ, or in
other words, that supp µ ⊆ u−1 (0) = Σp . For the other claim, first note that
PSH(Cn ) ∩ C(Cn )|Σp is dense in PSH(Σp ) ∩ C(Σ̄p ), this follows from the fact
that every u ∈ PSH(Σp ) ∩ C(Σ̄p ) extends to a plurisubharmonic function on
an open neighborhood of Σp in Cn . Hence, any v ∈ PSH(Σp ) ∩ C(Σ̄p ) can be
uniformly approximated by functions in PSH(Cn ) ∩ C(Cn ), so in particular,
by functions in PSH(P) ∩ C(P̄). Consequently, if v ∈ PSH(Σp ) ∩ C(Σ̄p ), and
µ ∈ Jpc (P̄), then
Z
v(p) ≤
v dµ ,
Σ̄p
Jpc (Σ̄p ).
or, in other words, µ ∈
Conversely, if u ∈ PSH(P) ∩ C(P̄), then
u|Σp ∈ PSH(Σp ) ∩ C(Σ̄p ). Hence if µ ∈ Jpc (Σ̄p ), then µ ∈ Jpc (P̄).
Let J and Σp be defined as in Proposition 2.3. From the fact that Σp ⊆ σJ
and the additional assumption that the polyhedron is non-degenerate we obtain
Corollary 2.4.
Corollary 2.4. If P is a non-degenerate analytic polyhedron in Cn , n ≥
2, then the following holds:
(1) if p ∈ ∂P and µ ∈ Jpc (P̄), then there exists an increasing multi-index
1 ≤ i1 < · · · < ik ≤ N of length k ∈ {1, . . . , n} such that the support of µ
is contained in σI ,
(2) if p ∈ Γn , then Jpc (P̄) = {δp }, where δp is the Dirac measure at p.
Let Ω ⊆ Cn be a bounded hyperconvex domain and let f : ∂Ω → R be a
continuous function. The Perron–Bremermann envelope is defined by
)
(
P Bf (z) = sup w(z) : w ∈ PSH(Ω), lim sup w(ζ) ≤ f (ξ) ∀ξ ∈ ∂Ω .
ζ→ξ
ζ∈Ω
In this setting P Bf is always plurisubharmonic, but not necessarily continuous.
In [14] Walsh proved that if
lim inf P Bf (z) = lim sup P Bf (z) = f (ξ)
z→ξ
z∈Ω
z→ξ
z∈Ω
for every ξ ∈ ∂Ω, then P Bf ∈ C(Ω̄). We will refer to this result as Walsh’s
theorem.
3. Proof of Theorem A. The outline of the proof of Theorem A is as
follows: We already know that (1), (4) and (5) are equivalent and therefore
we start by proving the equivalence between (2) and (3). Then we proceed by
proving that (2) implies (4) and finally that (4) implies (3).
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Proof of Theorem A. Assume that f is plurisubharmonic on the (n −
k)-dimensional complex manifold σI for every increasing multi-index I of length
k ∈ {1, . . . , n−1} and let d be an analytic disc embedded in ∂P, i.e., d : ∆ → P̄
is an injective, holomorphic function with d(∆) ⊆ ∂P. Proposition 2.1 implies
that there exists an increasing multi-index I0 of length k0 ∈ {1, . . . , n − 1}
such that d(∆) ⊆ σI0 and therefore (3) follows by Theorem 5.11 in [8]. For the
converse assume that (3) holds and let p ∈ σI , where I is a fixed increasing
multi-index I of length k ∈ {1, . . . , n − 1}. We now want to prove that f is
plurisubharmonic in a neighbourhood of p in σI . The functions fi1 , . . . , fik
which is a subset of the defining functions for P, can be used as a start of
a change of variables in a neighbourhood U of p in the following way: Let
w1 , . . . , wn−k be holomorphic functions on Cn−k such that
∂w1 ∧ · · · ∧ ∂fi1 ∧ · · · ∧ ∂fik ∧ · · · ∧ ∂wn−k 6= 0
on U . This is possible since P is non-degenerate by assumption. Moreover,
we will consider the functions w1 , . . . , wn−k as coordinates in Cn−k . Let L
be an arbitrary complex line through p in Cn−k and let d˜ : C → Cn−k be a
˜
holomorphic function with d(C)
= L. The function d : ∆ → P defined by
−1
˜
d = w ◦ d is an analytic disc with p ∈ d(∆) ⊆ σI and, by assumption, f ◦ d
is subharmonic. Hence, f is plurisubharmonic on σI , i.e., (2) holds.
Assume that (2) holds and let p ∈ ∂P and µ ∈ Jpc (P̄), then by Proposition 2.3 it follows that supp µ ⊆ Σp ⊆ σJ and µ ∈ Jpc (Σ̄p ) and therefore it
follows by assumption that
Z
Z
f dµ .
f dµ =
f (p) ≤
Σp
P̄
Consequently, (4) holds.
To complete the proof, we must now show that (4) implies (3). Let φ :
∆ → ∂P be an analytic disc and let ds denote the Lebesgue measure on ∂∆.
Then the push-forward of ds, φ∗ (ds), is a Jensen measure defined on Σ̄φ(0) with
barycenter φ(0) and therefore, by assumption and Proposition 2.3, it follows
that u satisfies (3).
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Received
October 24, 2007
Mid Sweden University
SE-851 70 Sundsvall
Sweden
e-mail : sam.lodin@miun.se
e-mail : frank.wikstrom@miun.se
Institute of Mathematics
Jagiellonian University
ul. Reymonta 4
30-059 Kraków
Poland
e-mail : Rafal.Czyz@im.uj.edu.pl